We confine our discussion on the ETC-BHM in the regime of polariton condensate. Under this situation, the mean-field treatment is suitable. To this end, we replace the operators in Hamiltonian (REF ) by their respective expectation values: \(P_\alpha \rightarrow \langle P_\alpha \rangle =\sqrt{N_\alpha (t)}e^{-i\theta _\alpha (t)}\)  (\(\alpha =1,2\) ) with \(N_\alpha (t)\)  and \(\theta _\alpha (t)\)  being the \(\alpha \) th polariton condensate's occupation number and phase, respectively [1]}, [2]}, [3]}. The equations of motion can be easily derived as:
\(\frac{d \xi }{d t} &=& 2N(T^+ +T^-\xi )\sqrt{1-\xi ^2}\sin \theta \,, \\\frac{d \theta }{d t} &=&\frac{N(V_1-V_2)}{2}+\frac{N(V_1+V_2-2U)}{2}\xi \nonumber \\&& -2N\frac{T^+\xi -T^-(1-2\xi ^2)}{\sqrt{1-\xi ^2}}\cos {\theta } \,.\) 
